circumbinary discs · 2020. 9. 24. · MNRAS 000,1{21(2020) Preprint 24 September 2020 Compiled...

MNRAS 000, 1–21 (2020) Preprint 24 September 2020 Compiled using MNRAS LATEX style file v3.0

The evolution of large cavities and disc eccentricity incircumbinary discs

Enrico Ragusa,1? Richard Alexander,1 Josh Calcino2, Kieran Hirsh3, Daniel J. Price41School of Physics and Astronomy, University of Leicester, Leicester, United Kingdom2School of Mathematics and Physics, The University of Queensland, QLD 4072, Australia3Univ Lyon, Univ Claude Bernard Lyon 1, ENS de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574,F-69230, Saint-Genis-Laval, France4 School of Physics & Astronomy, Monash University, Clayton, Victoria 3800, Australia

Accepted XXX. Received YYY; in original form ZZZ

ABSTRACTWe study the mutual evolution of the orbital properties of high mass ratio, circular,co-planar binaries and their surrounding discs, using 3D Smoothed Particle Hydro-dynamics simulations. We investigate the evolution of binary and disc eccentricity,cavity structure and the formation of orbiting azimuthal over-dense features in thedisc. Even with circular initial conditions, all discs with mass ratios q > 0.05 developeccentricity. We find that disc eccentricity grows abruptly after a relatively long time-scale (∼ 400–700 binary orbits), and is associated with a very small increase in thebinary eccentricity. When disc eccentricity grows, the cavity semi-major axis reachesvalues acav ≈ 3.5 abin. We also find that the disc eccentricity correlates linearly withthe cavity size. Viscosity and orbit crossing, appear to be responsible for halting thedisc eccentricity growth – eccentricity at the cavity edge in the range ecav ∼ 0.05–0.35.Our analysis shows that the current theoretical framework cannot fully explain theorigin of these evolutionary features when the binary is almost circular (ebin . 0.01);we speculate about alternative explanations. As previously observed, we find that thedisc develops an azimuthal over-dense feature in Keplerian motion at the edge of thecavity. A low contrast over-density still co-moves with the flow after 2000 binary orbits;such an over-density can in principle cause significant dust trapping, with importantconsequences for protoplanetary disc observations.

Key words: accretion discs – protoplanetary discs – hydrodynamics – planet-discinteractions – binaries

1 INTRODUCTION

Binaries are common in our Universe, and many phasesduring the formation and evolution of these binaries in-volve accretion discs. Their appearance in the electromag-netic spectrum depends on the nature of the objects com-posing the binary (black holes, stars, planets and moons)and the origin of the gaseous material surrounding them.Among these systems, protostellar/protoplanetary systems(star+star/planet) and black hole (BH) binaries (BH+BH)have recently attracted significant attention in the scientificcommunity.

On the one hand, protostellar/protoplanetary systemsare the outcome of the gravitational collapse of molecularcloud cores (for reviews, see Pringle 1989; Mac Low &Klessen 2004). Even when a binary system is formed, not all

? E-mail: [email protected]

the cloud material will land on the forming stars, and theremainder will form a disc around the binary. Furthermore,planet-disc interactions will be the result of planet formationfacilitated by the growth of dust grains. Planet-disc systemsare just binaries with extreme mass ratios.

Black hole binaries are expected to be found both inthe supermassive regime (SMBH binaries) in the gas-richcentres of galaxies powering AGN activity (Begelman et al.1980), and in the stellar regime (SBH binaries, the existenceof which has been confirmed by the detection of gravitationalwaves Abbott et al. 2016), marking the endpoint of the lifeof massive stars – outflows during the life of their stellarprogenitors throw gas in to the binary surrounds (de Mink& King 2017; Martin et al. 2018). Stellar BH binaries arealso expected to be found in the gas-rich central regions ofgalaxies (Stone et al. 2017; Bartos et al. 2017).

Despite the differences in physical scales between blackhole binaries and protostellar binary systems, the gas dy-

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namics is fundamentally the same, and the interaction be-tween binaries and discs appears to obey the same rules.

Conservation of angular momentum during infall on tothe binary forces the material to form a disc. The binaryexerts a tidal torque on the disc (Lin & Papaloizou 1979;Goldreich & Tremaine 1980), altering its structure by form-ing a gap (Crida et al. 2006; Duffell 2015; Kanagawa et al.2020) or, if the binary mass ratio is sufficiently high, a cavity(Cuadra et al. 2009; Shi et al. 2012; D’Orazio et al. 2013;Farris et al. 2014; Miranda et al. 2017). Vice versa, the discexerts a back-reaction torque on the binary causing evolu-tion of its orbital properties (migration, eccentricity evolu-tion) and also producing characteristic accretion patterns(Artymowicz & Lubow 1996; Günther & Kley 2002; Farriset al. 2014; Young et al. 2015; Ragusa et al. 2016; Muñozet al. 2019; Teyssandier & Lai 2019a).

The disc and the binary primarily exchange angularmomentum and energy at resonant locations (Goldreich &Tremaine 1979, 1980). A number of theoretical studies havebeen carried out investigating the effects of individual res-onances, in order to determine how they contribute to theevolution of the orbital properties (e.g. Artymowicz et al.1991; Goodman & Rafikov 2001; Rafikov 2002; Goldreich &Sari 2003).

Numerical studies have focused on the evolution ofbinary and disc parameters (e.g. Kley & Dirksen 2006;Paardekooper et al. 2010; Dunhill et al. 2013; Thun et al.2017; Kanagawa et al. 2018), probing the behaviour ofthe system for large secondary-to-primary mass ratios (e.g.Cuadra et al. 2009; Roedig et al. 2012; D’Orazio et al. 2013;Dunhill et al. 2015; Shi & Krolik 2015; D’Orazio et al. 2016;Muñoz et al. 2019, 2020), as the theory generally relies onthe assumption that the mass ratio of the binary, q, is � 1.

Some issues remain poorly understood, in particular thelong term evolution. The theoretical relationship betweenthe cavity truncation radius and binary properties (Arty-mowicz & Lubow 1994; Pichardo et al. 2005, 2008; Miranda& Lai 2015) appears to not be fully consistent with the nu-merical simulations on very long time-scales (Thun et al.2017; Ragusa et al. 2018), where in some cases binaries areobserved to carve larger cavities than are predicted theoret-ically. Recently, resonant theory was found in good agree-ment with numerical simulations by (Hirsh et al. 2020), butit failed to predict the cavity size for the circular, co-planarbinary case – on which we focus in this paper.

A number of numerical simulations starting with circu-lar discs and circular binaries show the growth of both bi-nary and disc eccentricity (e.g. Papaloizou et al. 2001; Kley& Dirksen 2006; D’Angelo et al. 2006; Dunhill et al. 2013;D’Orazio et al. 2016; Ragusa et al. 2018), even though aseed binary eccentricity e > 0 is required in order to excitethe eccentric Lindblad resonances which drive eccentricitygrowth (Ogilvie & Lubow 2003; Goldreich & Sari 2003). Fur-thermore, for high mass ratios, a crescent shaped over-densefeature orbiting at the edge of the cavity is likely to formfor almost any choice of disc parameters (Shi et al. 2012;Farris et al. 2014; Ragusa et al. 2016; Miranda et al. 2017;Ragusa et al. 2017). The physical mechanism(s) responsiblefor these features, and their long term evolution, are stillpoorly understood.

This last issue is of particular interest following the ob-servations performed by the Atacama Large Millimetre Ar-

ray, and other interferometers. These have imaged a numberof protostellar discs with cavities (sometimes referred to astransition discs) and prominent non-axisymmetric features(Tuthill et al. 2002; Andrews et al. 2011; Isella et al. 2013;van der Marel et al. 2016; Boehler et al. 2017; van der Marelet al. 2018; Casassus et al. 2018; Pinilla et al. 2018; van derMarel et al. 2019), whose origin is still being widely discussed(see Sec. 5 for a thorough discussion).

In this paper, we use a set of 3D Smoothed ParticleHydrodynamics (SPH) simulations to explore the mutualevolution of the binary, which is left free to evolve under theaction of the forces exerted by the disc, and disc orbital pa-rameters. We place particular emphasis on the evolution ofthe disc eccentricity and other disc orbital properties, aim-ing to explain the physical origin of the crescent shaped az-imuthal over-dense features in discs surrounding high massratio binaries, and understand the mutual interplay betweenthe binary and the evolution of disc eccentricity and cav-ity truncation radius. Long timescale 3D simulations (i.e.t & 1000 binary orbits) performing a similar analysis are notavailable in the literature. Three dimensional effects mightaffect the evolution of the eccentricity, as not allowing thematerial to access the vertical direction forcing it to movein the x-y 2D plane might spuriously increase the orbitaleccentricity.

We prescribe a simple locally isothermal equation ofstate, and we assume the binary and the disc lie on thesame plane. Other studies have been carried out to discussthe effects of misalignment between the disc and the binary(e.g. Bitsch et al. 2013b; Aly et al. 2015; Lubow et al. 2015;Nealon et al. 2018; Price et al. 2018b; Hirsh et al. 2020)and alternative prescriptions of the disc thermal structure(e.g. Baruteau & Masset 2008; Bitsch et al. 2013a; Beńıtez-Llambay et al. 2015).

We allow our simulations to evolve long enough to reachthe onset of a quasi-steady evolution. However, we note thatmost of the results presented in this paper focus on the tran-sition between the initial conditions and the quasi-steadystate, as we find that this phase lasts long enough to berelevant for the interpretation of the observations.

The paper is structured as follows: we begin with abroad introduction to resonant binary-disc interaction the-ory and how this affect the disc and binary evolution (Sec.1.1 and 1.2); In Sec. 2 we present our numerical simulations;Sec. 3 presents the results from the simulations; we discussthem in Sec. 4; in Sec. 5 we provide a detailed discussionabout the implications of our results in the context of pro-tostellar discs, we draw our conclusions in Sec. 6.

1.1 Resonant Binary-Disc Interaction

Resonant locations (or resonances) are regions in the discwhere the binary and the gas orbital frequency have an in-teger (or rational) ratio. At these locations the time-varyinggravitational potential of the binary excites density waves(Goldreich & Tremaine 1980). Waves carry angular momen-tum and energy that are transferred to the disc throughviscous dissipation or shocks (Goodman & Rafikov 2001).Resonances are identified by couples of integers (m, l) andcome in two broad types: co-rotation resonances, that are

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Evolution of disc cavities and eccentricity 3

located at

RC =( m

l

)2/3abin (1)

and Lindblad resonances, located at

RL =(

m ± 1l

)2/3abin, (2)

where ±1 depends on whether they are outer Lindblad res-onances (OLR) or inner Lindblad resonances (ILR), respec-tively. The efficiency of angular momentum transfer at agiven resonance depends on a number of factors (Goldre-ich & Sari 2003), such as the mass ratio of the binary, theeccentricity of the binary, the “type” of resonance and (forco-rotation resonances only) the disc vortensity gradient.

When the binary is circular, only resonances (m,m) areeffective as the intrinsic efficiency of each resonance scalesas e |m−l | , where e is the binary eccentricity (and not theexponential function). For this reason l = m resonances arecalled “circular” resonances. Circular corotation resonancesall fall at the co-orbital radius of the binary Rm,mC = abin andfor this reason they are also referred to as co-orbital reso-nances. When the binary is eccentric a new set of resonances,known as “eccentric” resonances, becomes effective.

The ratio between the exchange of angular momentumand energy is fixed by the properties of each resonance. Theoverall contribution of the interaction between the binaryand the disc at resonant locations determines the evolutionof the disc structure and binary orbital parameters. Thetorque exerted by the binary on the disc causes the formationof a gap, or if the mass ratio is sufficiently high (q > 0.04,D’Orazio et al. 2016)1, a cavity in the disc, and the onset ofdisc eccentricity (Lubow 1991; D’Angelo et al. 2006). Thedisc exerts a back reaction torque on the binary causing thebinary to migrate (change of semi-major axis) and changethe orbital eccentricity.

1.2 Mutual Evolution of Binary and Disc OrbitalProperties

All resonances lying in the circumbinary disc (i.e. outside thebinary orbit) cause inward migration of the binary, while in-ner resonances (within the binary orbit) promote outwardmigration. Different resonances in the disc provide differentcontributions to the binary eccentricity evolution (Goldreich& Sari 2003): outer circular Lindblad resonances (OCLR, i.e.RL > abin) and non-co-orbital eccentric Lindblad resonances(ELR with RL , abin) pump the eccentricity; while circularinner Lindblad resonances (ICLR, i.e. RL < abin), eccentriccorotation resonances (ECR) and co-orbital (i.e. ELR withRL = abin) resonances damp it. Furthermore, the evolutionof the disc density structure in the region of co-rotation res-onances is expected to cause them to saturate (Ogilvie &Lubow 2003), at which point these resonances become in-effective in their eccentricity damping action, allowing thebinary eccentricity to grow. The same ELRs expected topump the binary eccentricity are expected to pump the disc

1 We refer to this threshold value for the transition between gapand cavity as for mass ratios q > 0.04 no stable orbits aroundLagrange points L4 and L5 can be found (tadpole and horseshoe

orbits).

eccentricity, provided again that some initial disc eccentric-ity is present (Teyssandier & Ogilvie 2016).

Due to the absence of ELRs in discs surrounding circu-lar binaries, the evolution of the binary eccentricity in princi-ple should not be possible (Goldreich & Sari 2003). However,a number of numerical works have shown that it is possiblefor both the binary and the disc to increase their eccentric-ities, even in the absence of any initial “seed” binary or disceccentricity (Papaloizou et al. 2001; Dunhill et al. 2013). Itis important to note that this result is not surprising at all.The concept itself of circular Keplerian orbit by definitiondoes not imply the presence of a binary object at the centreof the system. Thus, initialising the velocities of fluid ele-ments around a binary using the Keplerian velocity alreadyprovides a small seed of orbital eccentricity for the disc. Fi-nally, we note that fixing the binary orbit throughout thelength of the simulation – as often done in previous works– breaks the conservation of angular momentum, possiblyleading to some spurious growth of the disc eccentricity.

In addition to the resonant interaction, secular interac-tions also affect the evolution of the disc and binary eccen-tricity (Miranda et al. 2017; Ragusa et al. 2018; Teyssandier& Lai 2019b) on long time-scales. Secular interaction is notexpected to provide long term growth or damping of thedisc eccentricity. Secular effects are instead responsible forperiodic oscillations of the eccentricity at fixed semi-majoraxis (exchange of angular momentum but not of energy).Secular interactions are also responsible for the precessionof the longitude of pericentre of both the binary and thedisc. Nevertheless, we note that there are some hints thatthe individual strengths of different oscillation modes (whichdepend on the disc-to-secondary mass ratio) appear to havesome role in determining the very long time-scale growth ofthe binary eccentricity (t & 105 binary orbits, Ragusa et al.2018).

2 NUMERICAL SIMULATIONS

We performed a set of numerical hydrodynamical simu-lations using the Smoothed Particle Hydrodynamics codephantom (Price et al. 2018a).

Our setup consists of two gravitationally bound massesM1 and M2 surrounded by a circumbinary disc (a cavity isalready excised when the simulation starts). These massesare modeled as sink particles, where gas particles can beaccreted (Bate et al. 1995). For numerical reasons we startall our simulations with Mtot = M1 +M2 = 1; we use differentbinary mass ratios q = M2/M1 that we will detail in Sec. 2.2.We initialize our binary on circular orbits with separationabin = 1. We use Rsink = 0.05 for both sinks. The sinks are freeto migrate due to their mutual gravitational interaction, andtheir interaction with the circumbinary disc. This enforcesconservation of angular momentum throughout the lengthof the simulation.

We use SPH artificial viscosity to model the physicalprocesses responsible for the angular momentum transferthrough the disc (as prescribed in Price et al. 2018a), thatresults in an equivalent Shakura & Sunyaev (1973) viscosity.We discuss the parameters we used for this purpose in Sec.2.2.

We allow our simulations evolve for t = 2000 torb, where

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4 E. Ragusa et al.

Table 1. Summary of the numerical simulations presented in thepaper. The reference name for each simulation contains a number,

that refers to the binary mass ratio, and a letter, that refers to

the disc properties used. Each simulation has been evolving forNorb = 2000 binary orbits.

Ref. q p H0/R0 αss Md/Mtot Rin

1A 0.01 1.5 0.05 5 × 10−3 5 × 10−3 2.02A 0.05 1.5 0.05 5 × 10−3 5 × 10−3 2.03A 0.075 1.5 0.05 5 × 10−3 5 × 10−3 2.04A 0.1 1.5 0.05 5 × 10−3 5 × 10−3 2.05A 0.2 1.5 0.05 5 × 10−3 5 × 10−3 2.06A 0.5 1.5 0.05 5 × 10−3 5 × 10−3 2.07A 0.7 1.5 0.05 5 × 10−3 5 × 10−3 2.08A 1.0 1.5 0.05 5 × 10−3 5 × 10−3 2.0

5C 0.2 1.5 0.10 5 × 10−3 5 × 10−3 2.05E 0.2 1.5 0.05 10−1 5 × 10−3 2.05Z 0.2 1.5 0.05 10−2 5 × 10−3 2.05N 0.2 1.5 0.03 5 × 10−3 5 × 10−3 2.05O 0.2 3 0.05 5 × 10−3 5 × 10−3 2.05P 0.2 0.25 0.05 5 × 10−3 5 × 10−3 2.05H 0.2 1.5 0.05 5 × 10−3 10−2 2.05A3.0 0.2 1.5 0.05 5 × 10−3 5 × 10−3 3.0

6A1.5 0.5 1.5 0.05 5 × 10−3 5 × 10−3 1.56A1.7 0.5 1.5 0.05 5 × 10−3 5 × 10−3 1.76A1.8 0.5 1.5 0.05 5 × 10−3 5 × 10−3 1.86A3.0 0.5 1.5 0.05 5 × 10−3 5 × 10−3 3.0

torb = 2π(GMtot/a3)−1/2 is the orbital time of the binary.We note that our choice of disc parameters implies a vis-cous time tν = 1.8 · 104–105 torb for radii R = 1–7. Evolvingthe system for such a long timescale is computationally in-tractable. However, we will see that that after 2000 torb allour discs reach a quasi-“steady” state, meaning that no fastvariations of the quantities examined throughout the paperare visible in our results at the end of our simulations (seealso the end of Sec. 5.1). We used Npart = 106 SPH particles.

2.1 Reference Case

In this section we introduce the disc setup that will be re-ferred to as the “A” setup throughout the paper (see all Ref.“A” in Table 1). The changes to the parameters used in thissetup will be detailed in the next section.

The initial circumbinary disc density profile in our sim-ulations extends from Rin = 2 to Rout = 7. For the inner edgeof the disc we follow the rule of thumb that tidally inducedcavities around circular binaries have Rin ≈ 2a (Pichardoet al. 2008). We also note that Rin lies in between the outer-most circular Lindblad resonance (m, l) = (1, 1) (OCLR, 2 : 1frequency commensurability, R1,1L = 1.59 abin) and the loca-tion of the outermost first order (m, l) = (2, 1) ELR (3 : 1frequency commensurability, R2,1L = 2.08 abin). We prescribea tapered power-law density profile of the type

Σ(R) = Σ0(

RR0

)−pexp

[−

(RRc

)2−p], (3)

where we use power-law index p = 1.5, reference radius R0 =Rin = 2 and tapering radius Rc = 5. We choose Σ0 in order tohave a disc-to-binary mass ratio Md/Mtot = 0.005. We use a

locally isothermal equation of state cs = cs,0(R/R0)−qcs withqcs = 0.25. We choose cs,0 in order to get a disc aspect-ratioH0/R0 = 0.05 at the reference radius R0.

Concerning the disc viscosity, we used an artificial vis-cosity parameter αAV = 0.2, β = 2 to prevent particle in-terpenetration, and allowed artificial viscosity to act also onreceding particles (as prescribed in Price et al. 2018a). Thisviscous setup results in an equivalent Shakura & Sunyaev(1973) viscous parameter αss = 0.005 (Lodato & Price 2010;Price et al. 2018a). We increase the value of αAV to obtainlarger values of αss in other setups.

2.2 Spanning the Parameter Space

In order to study how the system reacts to different phys-ical parameters we ran a large number of different simula-tions (See Table 1 for a list of the simulations). We varythe binary mass ratio q between the the following valuesq = {0.01, 0.05, 0.075, 0.1, 0.2, 0.5, 0.7, 1} (different numbers inthe “Ref”. column of Table 1 represent different values of q).In addition, we also vary some disc properties to test howthese affect the dynamics of the system. In each of them onesingle parameter is changed with respect to the disc referencecase “A” (different letters in the Ref. column of Table 1). Inparticular, in the case “5C” a thicker disc with H/R = 0.1is used; in the cases labelled as “5E” and “5Z” the disc ismore viscous than in the “5A” cases, using αss = 10−1 andαss = 10−2, respectively. The cases “5N, 5O, 5P, 5H, 5A3.0”use a thinner disc H/R = 0.03, a steeper initial density profilep = 3, a shallower density profile p = 0.2, a larger disc massand a different inner radius Rin = 3, respectively. Finally,in order to investigate the dependence on the initial condi-tions, we performed a set of simulations with q = 0.5 varyingthe inner disc radius Rin. In particular, simulations “6A1.5,6A1.7, 6A1.8, 6A3.0” have Rin = {1.5, 1.7, 1.8, 3.0} abin.

3 RESULTS

Figure 1 and 2 summarise how the surface density profile(vertically-integrated volume density) in the disc varies asa function of time in our reference simulations (simulationslabelled as “(1–8)A”) and for different initial inner disc radii(simulations 6A1.5, 6A1.7, 6A1.8, 6A3.0 in Table 1). Theseplots show the evolution of the surface density profile Σ(a, t)(colours, azimuthal average), as a function of the semi-majoraxis (x-axis) and time (y-axis). We stress here the impor-tance of producing density profiles using the semi-majoraxis as a space coordinate instead of radius (Teyssandier &Ogilvie 2017). When gas orbits in the disc become eccentric,plotting the density as a function of the radius is not ideal,as an element of material spans radii a(1 − e) ≤ R ≤ a(1 + e)along its orbit, and this makes it impossible to define theedge of the density profile precisely with a single value ofthe orbital radius.

All our simulations spend ∼ 400–700 torb in a “circular”steady state, maintaining their circular cavities and withoutaltering their size from the initial configuration. With the ex-ception of the case q = 0.01, for times t & 400 torb, an abruptgrowth of the semi-major axis and eccentricity of the cavityoccurs. Furthermore, for mass ratios q > 0.2, a prominentazimuthal over-density forms (see Fig. 3), which co-moves

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Figure 1. Disc surface density as a function of semi-major axis (x-axis) and time (y-axis) for “A” discs with different mass ratios (topleft to bottom right q = {0.01; 0.05; 0.1; 0.2; 0.5; 0.7; 1}, see Tab. 1). The magenta curve superimposed on the plot marks the location of the10% of the maximum value at each time (i.e. acav in Eq. (4)). Vertical lines in different colours mark the location of commensurabilitiesbetween the disc and binary orbital frequencies. The main ELRs responsible for eccentricity growth are located at the commensurabilities

1:2 (blue line), 1 : 3 (orange line) and 1 : 4 (green line). Note the abrupt transition in the cavity structure that takes place after ≈ 400binary orbits.

Figure 2. Disc surface density as a function of semi-major axis for a fixed mass ratio q = 0.5 but different disc initial radii ( simulations6A1.5, 6A1.7, 6A1.8, 6A3.0 see Tab. 1 for the simulations details with Rin = {1.5, 1.7, 1.8, 3.0}abin, respectively), in order to show that thetransition to the eccentric disc configuration occurs earlier when the inner disc is closer to the 1:2 resonance.

with the flow with Keplerian velocity. After this time-scale,the system moves to an “eccentric” configuration; the gasorbits consist of a set of nested ellipses with aligned pericen-tres and an eccentricity profile decreasing with radius (seetop panel of Fig. 4). The rigid precession of the disc longi-tude of pericentre Φd – i.e. the angle the pericentre formswith the positive x axis – always starts when the transitionto the eccentric configuration takes place. We note that thishappens because before that time the disc is circular, and itis therefore not possible to attribute any value to the lon-gitude of the pericentre. The binary also starts precessing,although at a much slower rate, as soon as the disc rigidprecession starts (see bottom panel of Fig. 5).

We note here that in a number of previous works theindividual masses of the binary are surrounded by circum-primary and circum-secondary discs (e.g. Farris et al. 2014;Ragusa et al. 2016; Miranda et al. 2017) – usually referredto as“circum-individual discs”or“mini”-discs. Given the rel-atively low disc viscosity and thickness in our simulations,if these discs form, the low rate at which the binary is fedwith the gas from the edge of the cavity makes them pro-gressively sparser, causing SPH numerical viscosity to growand triggering a positive feedback loop that leads to the dis-appearance of the circum-individual discs (see Sec. 3.3 forfurther discussion).

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6 E. Ragusa et al.

y

-5

0

5

q=0.01 q=0.05 q=0.075 q=0.1

y

x-5 0 5

-5

0

5

q=0.2

x-5 0 5

q=0.5

x-5 0 5

q=0.7

x-5 0 5 -7

-6

-5

-4

log

colu

mn

dens

ity

q=1.0

Figure 3. Gas surface density snapshots from simulations “A” – different panels show different binary mass ratios, as detailed in the

top right corner of each panel – after t ≈ 500 torb (apart from simulation 6A, with q = 0.50, which is taken at t = 750 torb, as the transitionto an eccentric configuration occurs at later times). The colour scale is logarithmic. An orbiting over-dense lump can be noticed in allsimulations with q ≥ 0.2.

3.1 Evolution of the Cavity Size

In order to provide a quantitative comparison, we definethe cavity size as the semi-major axis at which the valueof surface density azimuthal average reaches the 10% of themaximum of the profile at each time, such that

Σ(acav, t) ≡ 0.1 ×maxa[Σ(a, t)] . (4)

We show in the left panel of Fig. 6 the value of acav as afunction of time for our reference simulations (simulationslabelled as “A” in Tab. 1).

These density profiles were obtained by grouping gasparticles in semi-major axis bins, computing the semi-majoraxis of the i-th particle as

ai = −GmiMtot

2Ei, (5)

where Ei is the sum of the potential energy and kinetic en-ergy of the i-th particle and mi its mass. We note that sinceour estimate of ai depends on the total mechanical energy ofthe particle, the velocity corrections due to pressure effects(which we account for when initializing our discs) result inthe semi-major axis being slightly underestimated. We notethat this discrepancy for our purposes is negligible though,as it scales as ∆v2k ≈ (H/R)

2 . 1%.

3.2 Evolution of Disc Eccentricity

In order to quantify the disc eccentricity, we define a measureof the “global” disc eccentricity as follows. We compute thetotal disc angular momentum deficit (AMD) summing theindividual contribution of each particle in the disc domain

D = {R : 1.5 ≤ R ≤ 7} – a restriction of the disc domain isrequired as particles with R . 1.5 are no longer moving onKeplerian orbits. – as follows

AMDtot =∑i∈D

(Jcirc,i − Ji

), (6)

where the subscript i refers to the i-th particle, Jcirc,i =mi√

aiGMtot is the angular momentum of a particle of massmi and semi-major axis ai if it was on a circular orbit, Jiis the particle angular momentum. We then estimate the“total” eccentricity as

etot =

√√√√√2 AMDtot∑i∈D

Jcirc,i. (7)

We plot etot as a function of time for simulations “A” in theleft panel of Fig. 7. We remark that this definition provides aglobal estimate of the disc eccentricity and it is not meant togive a measure of the cavity eccentricity, which is generallyhigher.

Interestingly, the disc eccentricity grows rapidly until itreaches a maximum value. This value appears to depend onthe disc properties and binary mass ratio (a more thoroughdiscussion is provided in Sec. 4.1). As for the value of acav, wenote that, since the total eccentricity etot is computed usingEq. (6), the pressure velocity correction results in a small

spurious eccentricity etot,spur ∼√

1/2 × (H/R) even when thedisc is circular (such that etot,spur ∼ 3–5% in our simulations).

When in the eccentric configuration, discs always showan eccentricity profile that decreases with radius. As previ-ously mentioned, the disc longitude of pericentre points in

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Figure 4. Top panel: Azimuthally averaged eccentricity profile (colour) as a function of time (y-axis) and semi-major axis (x-axis); thered and magenta curves superimposed to the plot mark the location of the density maximum and location of the 10% of its value ateach time (i.e. acav in Eq. (4)). Bottom panel: Longitude of the pericentre (colour, azimuthal average) as a function of time (y-axis) andsemi-major axis (x-axis) from the simulations: 1A, 2A, 3A, 4A, 5A, 6A and 7A in Table 1 (q = 0.2 see Table 1). We show the maincharacterising features of the eccentricity evolution: growth of the eccentricity in the disc for t & 400 torb, eccentricity profile decreasingwith radius, and disc rigid precession of the pericentre longitude.

the same direction throughout the entire disc, precessing ona time-scale of the order of 100 torb, as shown in Fig. 4. Thisimplies that the disc behaves rigidly, as originally predictedby Teyssandier & Ogilvie (2016) and observed later numer-ically by Miranda et al. (2017) and Ragusa et al. (2018).

3.3 Evolution of the Binary Orbital Parameters

Since our simulations are performed with a “live” binary, theback reaction torque the disc exerts on the binary causes theevolution of the orbital properties of the binary.

We compute the binary semi-major axis abin as

abin = −GM1M2

2Ebin, (8)

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8 E. Ragusa et al.

Figure 5. Pericentre phase (y-axis) as a function of time (x-axis) for the disc at a = 4 (blue dots) and binary (orange dots) for the samesimulations in Fig. 4. The disc and the binary start precessing at the same time. The binary pericentre phase precesses at a slower ratethan the disc.

Figure 6. Left panel: Cavity size acav satisfying Eq. (4) as a function of time for the simulations: 1A, 2A, 3A, 4A, 5A, 6A and 7A inTable 1. We deliberately omit the case q = 0.01 in the left panel of the top row as acav as the algorithm to solve Eq. (4) in order to findacav of the maximum is not working properly for this case, being very close to the edge of the space domain we use for the analysis ofthe results (R = 1.5). Right panel: Same as left panel but for simulations: 5A, 5C, 5E, 5Z, 5N, 5O, 5P, 5H and 5A3.0 (see Table 1).

where Ebin is the binary mechanical energy

Ebin =12

M1v21 +

12

M2v22 −

GM1M2Rbin

, (9)

where Rbin = |R2 − R1 | is the physical distance between thetwo masses and v1 and v2 are velocities computed in the cen-tre of mass (CM) frame. We compute the binary eccentricityebin as

ebin =

√1 −

L2binµ2GMtotabin

, (10)

where Lbin is the total binary angular momentum, in the CMframe, and µ = M1M2M−1tot is the binary reduced mass.

Fig. 8 shows the evolution of the binary semi-major axisas a function of time. We note that for q ≥ 0.2 the evolu-tion of the semi-major axis is characterised by a temporary

increase in the migration rate of the binary. We will dis-cuss this effect later in Sec. 4.3. The lack of the circum-individual discs surrounding the binary (see the end of Sec.3) might impact the evolution of the binary – some recentworks showed that they might produce a positive torqueon the binary that lead to outward migration of the binary(Tang et al. 2017a; Muñoz et al. 2019; Moody et al. 2019;Duffell et al. 2019; Muñoz et al. 2020). However, Tiede et al.(2020) recently showed that migration still occurs inwardwhen the disc aspect-ratio is sufficiently small. More gen-erally, we note that conclusions regarding the evolution ofthe binary, such as exact migration or eccentricity growthrate, are beyond the scope of this paper. The presence of alive binary mainly allows us to capture secular oscillationsof the binary eccentricity, which might play a role in theevolution of the system, and informs us about the intensity

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Figure 7. Left panel: Total disc eccentricity etot in Eq. (7) as a function of time for the simulations: 1A, 2A, 3A, 4A, 5A, 6A and 7A inTable 1. Right panel: same as left panel but for the simulations: 5A, 5C, 5E, 5Z, 5N, 5O, 5P, 5H and 5A3.0 (see Table 1).

Figure 8. Binary semimajor axis abin in Eq. (8) as a function oftime for different mass ratios (simulations 1A, 2A, 3A, 4A, 5A,

6A, 7A, 8A in Table 1). We note that the migration rate increases

for simulations with q ≥ 0.2 when the cavity becomes eccentric.

of the binary-disc interaction – when the binary increasesits migration rate.

The left panel in Fig. 9 shows the evolution of the binaryeccentricity as a function of time for different binary massratios of our reference disc simulations. Secular oscillationsof the binary eccentricity can be seen for all cases madeexception for q = 0.01 and q = 1. The first is consistentwith the fact that both disc and binary remain circular forq = 0.01, and no oscillations can take place. The second, withthe fact that equal mass binaries (q = 1) are not expected toshow secular oscillations of the binary eccentricity (Mirandaet al. 2017).

Fig. 5 shows the evolution of the longitude of pericentreof the binary (orange lines) compared to the evolution of thedisc one (blue line, we remark that the disc precesses rigidly,i.e. same pericentre longitude at all radii). It is interestingto note that the precession rate is much lower in the bi-

nary than in the disc. The existence of two eigenfrequenciesfor the precession rate has been discussed and interpretedthrough a simplified toy-model in Ragusa et al. (2018).

3.4 Evolution of azimuthal over-dense features

As previously mentioned, when the transition to an “eccen-tric” configuration occurs, the disc develops a prominent az-imuthal over-density at the cavity edge with the shape of ahorseshoe. This feature can be seen in all our simulationswith mass ratio q > 0.2. Its initial contrast ratio grows withthe binary mass ratio, as shown in Fig. 10 where we evalu-ate the contrast ratio δφ of an over-dense feature by averag-ing the surface density in the surroundings of its maximumvalue and comparing it with the value at the opposite sideof the cavity. Each data point for the value of δφ in Fig. 10is obtained performing a moving average over a window of 5binary orbits. This makes the plot significantly less “noisy”,but it smooths away fluctuations in the contrast ratio thatmay occur on timescales shorter than 10 binary orbits.

In order to better capture the orbiting/non-orbiting na-ture of such features, we introduce also two additional fig-ures. First, Fig. 11 shows 15 snapshots towards the end ofsimulation 6A (q = 0.5, times shown are t = 1985–2000torb,one per binary orbit). Second, Fig. 12 shows a Lomb-Scargleperiodogram of the accretion rate on to the binary through-out the length of simulation 6A. Frequencies are on the x-axis (in units of (torb)−1), colours code the powers of differ-ent frequencies and times are on the y-axis. Horizontal lines(t = const) in this plot represent the periodogram of the ac-cretion rate on a window of 30 binary orbits, at a fixed timeof the simulation.

From now on, when relevant, we will distinguish amongazimuthal over-dense features referring to them as “orbitingover-dense lumps”, when the feature moves with Keplerianmotion at the edge of the cavity, and “eccentric traffic jam”,for non orbiting features. We discuss the formation mecha-nism of these features in detail in Sec. 4.3.

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10 E. Ragusa et al.

Figure 9. Left panel: binary eccentricity ebin in Eq. (10) as a function of time for simulations 1A–8A (left panel) and for simulationswith mass ratio q = 0.2 and different disc parameters (right panel).

3.5 Results for Different Disc Parameters

Here we present a second set of simulations we performedfor a fixed mass ratio q = 0.2 while varying some of thedisc parameters. Right panels in Fig. 6 and 7 show the timeevolution of the cavity size (left panel) and “total” disc ec-centricity (right panel) using Eq. (7) for the simulations: 5A,5C, 5E, 5Z, 5N, 5O, 5P, 5H and 5A3.0 in Tab. 1. We first notethat interestingly the case 5A3.0 suggests that the transitionto the “eccentric” disc configuration takes place only whenthe still circular cavity edge reaches a minimum separationfrom the binary. Indeed, simulation 5A3.0, being initialisedwith a larger cavity (Rin = 3), shows a delay in the growth ofthe disc eccentricity, probably due to the need for the disc tospread viscously until it reaches some resonant location. Theslope of the density profile (5O and 5P), the disc mass (5H),and small changes in the disc viscosity (5Z) are not causingsignificant differences in the evolution of the disc eccentric-ity. An increased disc thickness (sim 5C), besides opposingthe opening of a cavity due to the stronger pressure gradientat the cavity edge, increases the disc viscosity ν, since it isparametrised using the Shakura & Sunyaev (1973) prescrip-tion. This provides a faster spread of the disc disc towardsthe resonant location and transition to the disc eccentricconfiguration at earlier times, even though the maximumdisc eccentricity is lower than in the reference case. Explic-itly increasing the disc viscosity through the αss parameter(sim 5E) produces the same effect. Consistent with this sce-nario, a reduction in the disc thickness (sim 5N) shows thatthe disc transition to the eccentric configuration occurs atlater times. In this last case the final value of the disc ec-centricity is higher than in the reference case 5A. The rightpanel in Fig. 9 shows how different disc parameters affectthe evolution of the binary eccentricity.

Simulations 5A3.0, 6A1.5, 6A1.7, 6A1.8 and 6A3.0 allshare the same disc properties of “A” discs, with the onlydifference that their initial inner truncation radius is set toRin = {1.5, 1.7, 1.8, 3.0} abin according to their reference label,as outlined in Table 1. The surface density evolution fromthese simulations appear in Fig. 2, they will be discussedfurther in Sec. 4.3.

4 DISCUSSION

The results presented in the previous sections hint at a num-ber of interesting features in the evolution of eccentric discs.We note that items i-iii below have been previously discussedin the literature, items iv-viii, to our knowledge, did not re-ceive the same attention and will be subject of a deeperdiscussion.

(i) All discs with sufficiently large binary mass ratios(which here appears to be q & 0.05) become eccentric, con-sistently with what previously found (D’Orazio et al. 2016;Ragusa et al. 2017; Muñoz & Lithwick 2020). Previous stud-ies have shown that binaries with mass ratios q < 0.05 mayexcite the eccentricity of the circumbinary disc (D’Angeloet al. 2006; Kley & Dirksen 2006; Teyssandier & Ogilvie2017). Nevertheless, in this work we mainly refer to theabrupt growth of the disc eccentricity that occurs in mostof our simulations after an initial “circular phase” that lastsfor ∼ 400 − 700 orbits (see Fig. 1 and top panel of 4 – seealso 14 below).

(ii) Eccentric discs undergo rigid longitude of pericentreprecession (the pericentre of the eccentric disc orbits re-mains aligned throughout the entire disc, see bottom panelof Fig. 4). The precession rate of the disc is independent fromthat of the binary, which precesses at a much slower rate(Fig. 5). This had been previously found numerically (Mac-Fadyen & Milosavljević 2008; Miranda et al. 2017; Thunet al. 2017; Ragusa et al. 2018) and discussed theoreticallyby Teyssandier & Ogilvie (2016), and recently by Muñoz &Lithwick (2020). The physical explanation of the origin ofthis pericentre alignment is that the clustering of eccentricorbits at the apocentre is “pinching” (Dermott & Murray1980)2 together all the elliptic orbits, preventing them fromprecessing differentially.

(iii) As noted above, a prominent orbiting over-denselump develops for mass ratios q > 0.2 that is co-movingwith the gas (i.e. not an eccentric “traffic jam”) as shown in

2 We note that in (Dermott & Murray 1980) the“pinching”occursat the pericentre as the eccentricity profile has a positive gradient.

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Figure 10. Density azimuthal contrast ratio of as a function of time for simulations “A” with different mass ratios (left panel) and

simulations 6A1.5, 6A1.7, 6A1.8 and 6A3.0 (right panel) (see Tab. 1). We note that when the cavity size grows while the disc becomeseccentric, for q ≥ 0.2 the disc develops a pronounced azimuthal asymmetry, that progressively decays to a density contrast of ∼ 4 after∼ 1000 binary orbits, consistent with that expected from an eccentric “traffic jam”. Larger initial inner disc radii Rin postpone the timethe disc transitions eccentric configuration and reduces the maximum contrast ratio the over-density can achieve.

y

-5

0

5 t=1985 t=1986 t=1987 t=1988 t=1989

y

-5

0

5 t=1990 t=1991 t=1992 t=1993 t=1994

y

x-5 0 5

-5

0

5 t=1995

x-5 0 5

t=1996

x-5 0 5

t=1997

x-5 0 5

t=1998

x-5 0 5

0

5×10-5

1×10-4

colu

mn

dens

ity []

t=1999

Figure 11. Surface density map of simulation 6A for times t = 1985–1999 torb, using a linear colour scale. The plot is meant to show that,besides the “eccentric” over-dense feature at the cavity apocentre, a periodic (≈ 7–8 torb) variation of the density at cavity pericentre stilloccurs at the end of the simulation. The reader will notice that, besides the higher contrast non-orbiting feature at the cavity apocentre(North of each snapshot), at the cavity pericentre (South) the surface density varies by a factor ≈ 1.5–2 every ≈ 7–8 torb. This suggeststhat a low contrast over-density (δφ ∼ 1.5–2) is still orbiting, co-moving with the flow, at the edge of the cavity. See also Fig. 12.

Fig. 14. This feature has been observed in previous works(e.g. Farris et al. 2014; Miranda et al. 2017; Ragusa et al.2017) and is referred to as “over-dense lump” or “horseshoefeature” in the black hole and protoplanetary disc commu-nity, respectively. A discussion about its formation mecha-nism has been provided by Shi et al. (2012). However, manyaspects regarding formation and evolution of such featuresremain unclear – see Sec. 4.3 for further discussion.

(iv) In all cases the binary gains a small amount of ec-centricity before entering the “eccentric cavity” phase –ebin ∼ 0.001 − 0.007 in most cases. Two cases, which respec-tively used a more massive and a thinner disc than the ref-erence case (simulations 5H, larger disc mass, and 5N, lowerdisc thickness, in Table 1), rapidly reach ebin ∼ 0.01 and keepgrowing. This behaviour is consistent with previous studies(Dunhill et al. 2013; Ragusa et al. 2018), which found that a

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12 E. Ragusa et al.

Figure 12. Periodogram of the accretion rate ÛM on to the binary(frequency is on the x-axis and powers are coded with different

colours) at different times (y-axis). The x-axis report frequenciesin units of (torb)−1, so that the vertical line centred at frequencyt−1 = 0.5 (torb)−1 for the first 500 binary orbits and t−1 = 1 (torb)−1from t & 800 torb indicates that the binary shows a modulation ofthe accretion rate once every two binary orbits and once every

orbit, respectively. On the top of that, a slower modulation with

frequency t−1 = 0.12–0.13 (torb)−1 appears as soon as the orbitingover-dense lump forms (t & 800 torb). Such an accretion feature islinked to the presence the orbiting over-density which makes a

close passage at the cavity pericentre every 7–8 torb (as also shownin Fig. 11). The presence of such accretion feature at late times

implies that an orbiting over-dense lump of material is still present

at the end of the simulation.

larger disc-to-binary mass ratio qd = Md/(M1 +M2) (sim 5H,qd = 0.01) leads to a larger binary eccentricity. The highervalue of the binary eccentricity associated to a thinner disc(sim 5N, H/R = 0.03) is consistent with a reduction of theresonance width for lower values of H/R, which provides astronger Lindblad torque on the binary (Meyer-Vernet &Sicardy 1987). See Eq.s (21) and (22) in Goldreich & Sari(2003) for the dependence of binary torque on both disc massand resonance width.

(v) The duration of the initial phase during which thedisc remains circular depends on the initial radius of thedisc, suggesting that the disc spreads viscously and then en-counters resonances. Our results suggest that the resonanceslocated at the 1:2 frequency commensurability play a role inexplaining the evolution of the disc eccentricity we observe,as previously suggested by D’Angelo et al. (2006). We willdiscuss further about the role of different resonances (or al-ternative non-resonant mechanisms) for the growth of thecavity eccentricity in Sec. 4.3. We will see that it is hard tointerpret the results within the theoretical framework cur-rently available in the literature, posing the basis for possiblefuture developments of the theory.

(vi) In all the simulations, the disc eccentricity stopsgrowing when it reaches a maximum value. For q > 0.5, thedisc eccentricity appears to saturate at a maximum valuewhich is independent of the binary mass ratio – inner edgeof the cavity ed(acav) ∼ 0.5, “total” eccentricity ed,tot ∼ 0.25.For q < 0.5, this maximum value scales with the disc viscos-ity (see Fig. 7). We discuss this further in Sec. 4.3.

(vii) When discs become eccentric, they appear to havelarger cavities than when they are circular: the cavity semi-major axis becomes up to 3.5 times the binary separation,whereas during the start of the simulations the inner edgesof the disc all remain at ∼ 2 binary separation for ∼ 400binary orbits (see Fig. 1 and left panel of Fig. 6). There isa strong correlation between the cavity eccentricity and itssize, as shown in the left panel of Fig. 13. We will discussthis aspect further in Sec. 4.2.

(viii) The radius of the cavity pericentre, the evolution ofwhich is shown in the right panel of Fig. 13, remains ap-proximately constant at Rd,peri = acav(1− ed,tot) . 2 through-out the simulation. This is consistent with the correlationfound between acav and ed,tot. This suggests that the mini-mum separation of gas particles from the binary is fixed for agiven mass ratio, and growth of the disc eccentricity (at fixedpericentre) therefore results in corresponding growth of thecavity semi-major axis – see Sec. 4.2 for further discussion.

In the following sections we interpret these resultswithin the existing theoretical framework. We also speculateabout possible interpretations of some results that cannot beexplained with our current understanding of resonant andnon-resonant binary-disc interaction, hinting at the direc-tion that further theoretical studies should take to confirmthe interpretation of our numerical results.

4.1 Evolution of the cavity eccentricity

In Sec. 1.1 we discussed the role resonant interaction is ex-pected to play for the evolution of both binary and disc ec-centricity. Non-resonant mechanisms might also play a role.We identify four possible sources of the eccentricity growth.

(i) Resonance (m, l) = (3, 2) ELR located at the 1:2 binary-to-disc orbital frequency commensurability (RL = 1.59 abin).This resonance is expected to pump the disc eccentricity(Goldreich & Sari 2003); its role in the evolution of the disceccentricity has been previously discussed by D’Angelo et al.(2006). This resonance requires the binary eccentricity to beebin , 0 to be effective, which, despite small values of binaryeccentricity are excited, is the case for our simulations.

(ii) At the same location as the {m, l} = {3, 2} ELR(RL = 1.59 abin), lies the (m, l) = (1, 1) OCLR, which is ex-pected to pump the disc eccentricity as well. This resonanceis effective also for circular binaries. We note that OCLRsare the only available resonances that can increase the discand binary eccentricity if the binary is fixed on a circularorbit (MacFadyen & Milosavljević 2008).

(iii) Resonance {m, l} = {2, 1} ELR at the 1:3 frequencycommensurability (R2,1 = 2.08 × abin) has also been previ-ously suggested to play a role in eccentricity evolution byPapaloizou et al. (2001).

(iv) The lack of stable closed orbits around Lagrangepoints L4 and L5 for binary mass ratios q > 0.04 has beendiscussed to be possibly causing the growth of the disc ec-centricity (D’Orazio et al. 2016); this mechanism is non-resonant.

(v) Impact of gaseous streams from the disc cavity peri-centre hitting the opposite edge of the cavity wall (Shi et al.2012; D’Orazio et al. 2013). This mechanism is non-resonant.

The disc eccentricity has been shown to increase expo-

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Figure 13. Left panel: Correlation between disc eccentricity (y-axis) and cavity semi-major axis (x-axis) throughout the entire length of

the simulation for different “A” discs. Right panel: the cavity pericentre radius as a function of time, computed using Rd,peri = acav(1−ed, tot)for different “A” discs. As in the left panel of Fig. 6 we deliberately omit the case q = 0.01.

nentially when a small eccentricity seed in the disc is present(see Eq. (14) in Teyssandier & Ogilvie 2016). Thus, as soonas the disc and/or the binary have a small fluctuation intheir orbital eccentricity, if the disc covers an OCLR/ELRlocation, the eccentricity increases rapidly until some otherphysical mechanism limits its growth. Non-resonant growthof the disc eccentricity has not been quantitatively discussedin the literature. However, in essence, as soon as the gasspreads towards the co-orbital region it will be forced tomove on non-closed orbits, perturbing the circularity of thecavity. As for resonant mechanisms, non-resonant eccentric-ity growth stops when eccentricity damping through somesecondary mechanism becomes dominant.

As soon as the simulation starts, the disc viscouslyspreads inward from its initial radius, without growing its ec-centricity at all. Then a three-lobed structure appears (seeFig. 14). Immediately after the appearance of this three-lobed structure, the disc eccentricity rapidly increases to-ward its maximum value.

Our simulations were started with Rin = 2 in order todeliberately cover the 1:3, {m, l} = {2, 1}, binary-disc or-bital frequency commensurability, which is located at RL,21 =2.08 abin. Given the delay in the growth of the eccentricity,we can exclude this resonance as being the main contrib-utor to the abrupt growth. Resonances {m, l} = {3, 2} and{m, l} = {1, 1}, located at the 1:2 binary-disc orbital fre-quency commensurability (RL,32 = 1.59 abin) appear to bebetter candidates (see also D’Angelo et al. 2006; MacFadyen& Milosavljević 2008; Miranda et al. 2017). The right panelof Fig. 10 shows that reducing the disc inner radius Rin atthe beginning of the simulation moves forward the abruptgrowth of the disc eccentricity and cavity size (producingthe high contrast ratio showed in that plot); the growth ofeccentricity starts immediately when Rin = 1.5 < RL,32 forsimulation 6A1.5. Resonance {m, l} = {1, 1}, being a circu-lar resonance, appears to be the strongest resonance amongthose proposed. However, the formation of the three-lobedstructure suggests an m = 3 resonance is effective, implyingthat also {m, l} = {3, 2} ELR resonance might be playing arole, despite the binary eccentricity being small ebin . 0.01.

Non-resonant eccentricity growth cannot be excluded, but itis hard to justify the m = 3 spiral in that framework.

We note that for binaries with mass ratio q = 1 theOCLR resonance {m, l} = {1, 1} is not effective. MacFadyen& Milosavljević (2008) showed that the OCLR resonance{m, l} = {2, 2} located ad the 3:2 binary-to-disc frequencycommensurability (RL,22 = 1.31 abin) is effective for thegrowth of the eccentricity for this specific case. However, it isnot clear whether in our simulations some material reachesthat separation before the abrupt disc eccentricity growthstarts. More generally, our simulation with q = 1 (simula-tion 8A) does not show sufficient evolution of the binaryeccentricity for ELRs to be effective, making the growth ofthe disc eccentricity for the q = 1 case hard to justify withinour current understanding of resonances. In Sec. 4.2 we willspeculate about the possible growth of the intensity of ELRsto solve this and other issues that will arise when discussingthe cavity size.

The evolution of the disc eccentricity is ruled by com-peting effects that damp or pump the eccentricity. We men-tioned in Sec. 1.1 that co-rotation resonances damp the ec-centricity, but they are expected to saturate and lose theircircularising effect. Viscous dissipation and eccentric orbitintersection, which occurs when the disc eccentricity gra-dient is sufficiently steep to satisfy the following criterion(Dermott & Murray 1980)

a(

deda

)& 1, (11)

then become the main mechanisms acting to damp the disceccentricity. Nested elliptical orbits with eccentricity de-creasing with radius are expected to be subject to frictionwhich becomes stronger depending on the eccentricity gra-dient.

Viscous dissipation occurs when the eccentricity pump-ing effect is progressively balanced by the damping effectprovided by viscosity. The criterion for orbit intersection isinstead a physical limit, beyond which strong shocks rapidlydamp the eccentricity. This explains the maximum valuesthe disc eccentricity can reach in our simulations. With refer-

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14 E. Ragusa et al.

ence to Fig. 7, in our reference simulations (“A” simulationsin Tab. 1) cases with q < 0.5 progressively grow their eccen-tricity until viscous circularisation balances the effect of theELRs. For larger mass ratios (q ≥ 0.5), ELRs can in princi-ple pump more eccentricity in the disc, but orbit intersectionprevents its further growth, causing the eccentricity to sat-urate to the same maximum value for all q ≥ 0.5. Fig. 16shows a comparison between the values of Eq. (11) through-out the disc. We see clearly that the region at the edge ofthe cavity fully satisfies the criterion for orbit intersectionfor q = 0.7 (sim 7A), but not for q = 0.2 (sim 5N). Despitereaching very similar maximum values of disc eccentricity,the case 5N, with a thinner disc, has a lower value of discviscosity and thus the viscous damping is weakened, allow-ing the eccentricity to grow to larger values, with respect tothe reference q = 0.2 case (sim 5A). However the steepnessof the eccentricity profile does not appear to be high enoughto provide orbit intersection.

Orbits do not intersect for shallow eccentricity profilesfor any value of the eccentricity (see Fig. 16). The maximumvalue the eccentricity can reach is thus determined by twomechanisms. When the right resonances are excited, the bi-nary pushes the disc to become eccentric, while the viscousdissipation in the disc tends to circularize the orbits. Whenthese two processes balance, the eccentricity stops evolving.

4.2 Cavity size: strengthening of EccentricResonances or non-Resonant Truncation?

If small binary eccentricities as those we observe in our sim-ulations can in principle activate ELRs, causing them toproduce typical density features – as the m=3 spiral rightbefore the onset of the eccentricity growth – it is very hardto believe they have sufficient strength to truncate the disc.

In our simulations we observe an initial growth of the bi-nary eccentricity (see Fig. 9), which could in principle causethe strength of ELRs to grow.

However, the values of binary orbital eccentricity thatare excited in our simulations are in most cases ebin < 0.01.Our current understanding of ELRs tells us that for suchsmall values of binary eccentricity, resonances cannot over-come the viscous forces in the disc to open a large cavity(Artymowicz & Lubow 1994). We show this in Fig. 17, wherewe provide an estimate of the intrinsic strength of ELRswhose location in the disc is consistent with the size of thecavity, and then compare it to a criterion for a cavity to beopened by that resonance (Artymowicz & Lubow 1994).

OCLRs cannot be invoked to explain disc truncation atradii R > RL,11 = 1.59 abin, being the {m, l} = {1, 1} the out-ermost circular Lindblad resonance. Given the low strengthof ELRs in our simulations, we cannot invoke resonant trun-cation to explain cavities as large as acav ≈ 4 abin shown inFig 6.

We here speculate about two possible scenarios that canbe responsible of the depletion of such large cavities.

First, the intrinsic strength of ELRs also depends onthe disc eccentricity, instead of being exclusively related tothe binary one. This possibility would set the basics for anew physical mechanism producing the unstable growth ofcavity size, which relates with the disc eccentricity since thethe higher its value is, the stronger the outer ELRs are.This speculative scenario is supported by calculations that

use fixed binaries (i.e., not allowed to change their orbitalparameters, as we allow here) showing an evolution of thecavity structure in terms of size and eccentricity beyond thelocation of the outermost circular Lindblad resonance (e.g.Kley & Dirksen 2006; Shi et al. 2012; Farris et al. 2014;D’Orazio et al. 2016), implying that having ebin , 0, is nota fundamental requirement in order to activate ELRs.

Second, a non-resonant mechanism sets the cavity size(Papaloizou & Pringle 1977; Rudak & Paczynski 1981;Pichardo et al. 2005, 2008). In this interpretation disc trun-cation takes place at the innermost separation from the bi-nary where gas orbits are not anymore “invariant loops”(Pichardo et al. 2005, 2008), implying that gas particle orbitsare “intersecting”, dissipating the orbital energy and clear-ing the cavity region – which is analogous to what happenswhen the eccentricity gradient exceeds the “orbit intersec-tion” threshold (see above Eq. (11) in Sec. 4.1).

Previous studies which considered the dependence ofthe truncation radius of the cavity on the binary propertiesassumed circular orbits in the disc (Pichardo et al. 2005,2008). We speculate that non-intersecting orbits of the gasare expected up to a minimum separation from the binary,and this sets the pericentre radius of the cavity edge. Thisis supported by the behaviour of the cavity pericentre, plot-ted in the right panel Fig. 13: where the pericentre radiusdepends on the binary eccentricity and mass ratio, but re-mains roughly constant for all the mass ratios throughoutthe simulation.

If the pericentre radius Rd,peri of the cavity is fixed non-resonantly, when the disc eccentricity grows, it will cause thegrowth of the cavity semi-major axis. The correlation be-tween the cavity size and disc eccentricity shown in Fig. 13supports this scenario. The relationship between the disc ec-centricity and innermost non-intersecting orbit has not beenestablished yet, and will be subject to future studies.

Completing this second scenario, we note that if thenon-resonant eccentricity growth scenario introduced in Sec.4.1 is effective, the mechanism we describe would be com-pletely non-resonant.

However, we note that we are not able to verify this in-terpretation without substantial further development of thetheoretical framework of resonant and non-resonant binary-disc interaction for eccentric discs. Indeed, we are not awareof any previous work suggesting or directly studying theseeffects.

4.3 Formation of the azimuthal over-density

One of the most interesting features that arises from thesesimulations is the formation of a well defined azimuthalasymmetry in the density field (see right panel of Fig. 14) inall simulations with binary mass ratio q > 0.2. This featureis often referred to as a “horseshoe” in the protoplanetarydisc community, due to its shape, or “over-dense lump” inthe black hole community. It has been seen in a number ofprevious studies (Shi et al. 2012; Farris et al. 2014; Ragusaet al. 2016, 2017; Miranda et al. 2017; Calcino et al. 2019).In order to evaluate the intensity of the asymmetry we de-fine the contrast ratio δφ as the ratio between the density inthe azimuthal feature, and the density at the opposite sideof the cavity.

If the disc eccentricity profile has a negative radial gra-

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y

x-5 0 5

-5

0

5

x-5 0 5 -7

-6

-5

-4

log

colu

mn

dens

ity

Figure 14. Snapshots of the case 6A (q = 0.7) after t = 330 torb (left panel) and t = 450 torb (right panel). A Three lobed cavity (leftpanel) marks transition from a “small” circular cavity to a “large” eccentric cavity.

Figure 15. Disc eccentricity as a function of time (y-axis) and semi-major axis (x-axis) for the two cases 1N (left panel) and 4A (rightpanel), see Table 1. A higher eccentricity can be achieved in the case 1N than in case 4A since orbits are not crossing (see also Fig. 16).

dient, it is expected to form an non-orbiting azimuthal over-density with contrast ratio δφ ≈ 3– 4. This is referred to asthe “eccentric feature” or “traffic jam”, and caused by theclustering of orbits at their apocentres (Ataiee et al. 2013;Teyssandier & Ogilvie 2016; Thun et al. 2017). This feature(we refer to it as “eccentric traffic jam”) is fixed at the apoc-entre of the cavity, and moves only because of the precessionof the longitude of pericentre of the cavity.

The over-density visible in Fig. 14 (we refer to it as“orbiting over-dense lump”) not only moves around the edgeof the cavity with Keplerian motion, but reaches a contrastratio δφ ≥ 10, larger than the typical values in “traffic jams”.

Tidal streams being thrown from the cavity pericen-

tre against its opposite edge have been discussed to causethe formation of the feature (Shi et al. 2012; D’Orazio et al.2013). Consistent with this picture, from our work it emergesthat one of the key elements for the formation of a high con-trast ratio over-density is the fast outward motion of the gaswhen the cavity progressively increases in size. Moving out-wards, the gas first produces an over-dense ring of material,which then evolves into an azimuthal structure.

Ragusa et al. (2017) found a threshold value for theformation of high contrast over-densities of q > 0.05. Simu-lations in that work used discs that initially extended up toRin = 1.5 < RL,11, making that result consistent with whatwe found in this paper. Even though no whirling motion is

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16 E. Ragusa et al.

Figure 16. Colour-plot of the quantity in Eq. (11) as a function of time (y-axis) and semi-major axis (x-axis) for the cases 5N (left

panel) and 6A (right panel), as in Table 1. When a de/da > 1 eccentric orbits are expected to cross, suppressing further eccentricitygrowth in the disc, as shown in Fig. 15 and 7 where the case 5N clearly reaches a larger value of maximum eccentricity.

Figure 17. Intensity of individual resonances using the r.h.s. of

Eq. (16) by Artymowicz & Lubow (1994) (y-axis) vs their loca-tion in the disc (x-axis). The dashed line represents the viscous

threshold in their intensity in order to create a depletion in thedisc surface density. Each intensity is computed at different times

using the orbital properties of the binary (crosses). Star markersare instead computed using the disc eccentricity. We use differentmarker colours to indicate the order of ELRs: namely, purple are

first order (m = l), cyan second order (m = l+1), olive third order(m = l + 2), coral fourth order (m = l + 3). We note that this is notmeant to prove our claims, but it simply shows that if the disc

eccentricity affects the intensity of resonances in the same way asthe binary, a growth of the cavity size is reasonable to occur upto the 1:5 resonant location.

present in orbiting over-dense lumps, Hammer et al. (2017)noted that RWI vortices are weaker when planets inducingthem appear in the simulation slowly increasing their mass;similarly, this leads to a slower buildup of material at thegap/cavity edge.

4.3.1 Evolution and life expectation of the azimuthalover-dense feature

This qualitative picture described above appears to be con-firmed by the evolution of the contrast ratio. Fig. 10 showsthe contrast ratio of the azimuthal over-density, δφ, as afunction of time for our “A” reference cases and for four dif-ferent choices of initial Rin of the simulation (simulations6A1.5, 6A1.7, 6A1.8 and 6A3.0).

The contrast ratio of the azimuthal over-density growswhen the cavity size starts growing, reaching a peak whenthe cavity size reaches its maximum value. When the mate-rial stops moving outward, the over-density stops growing.After its peak, the contrast ratio of the over-density pro-gressively decreases – probably due to viscous dissipation –until it appears to stabilise at a value of ≈ 4 for q ≥ 0.2,which are characterised by the maximum eccentricity gradi-ent being fixed by orbit-crossing limit, Eq. (11). This finalvalue of the contrast ratio corresponds to the contrast ra-tio of the slowly precessing “eccentric traffic jam” structurethat forms at the cavity apocentre following the growth ofthe disc eccentricity.

Initializing the disc with a larger inner radius Rin leadsto later growth of the cavity size and eccentricity (Fig. 2).Since the amount of material that viscously spreads inwardis less than that present in a simulation starting with thedisc extending to smaller inner radii, the amount of mate-rial pushed outward when the cavity becomes eccentric andincreases its size is lower, resulting a less pronounced over-dense feature. When starting the simulation with Rin = 3(simulation 6A3.0) the system seem to directly form an “ec-centric traffic jam” feature.

Despite the “eccentric traffic jam” becoming the domi-nant over-density at late times, a feature with a contrast ra-tio of ∼ 1.5–2 keeps orbiting at the cavity edge, as shown inFig. 11. This result is consistent with what previously foundby Miranda et al. (2017), where the authors found that anorbiting over-density with contrast δφ ∼ 2–3 is found to bepresent after 6000 binary orbits, with a viscosity 10 times

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higher than the one used in the present work3. Such featuremakes a close passage to the binary when it reaches the cav-ity pericentre, i.e. every ∼ 7–8 binary orbits. This boosts theaccretion rate with the same periodicity (see Fig. 12) as pre-viously found in a number of work (e.g. Cuadra et al. 2009;D’Orazio et al. 2013; Farris et al. 2014; Ragusa et al. 2016;Miranda et al. 2017).

We note that in our analysis (not shown here to reducethe paper length) also simulation 6A3.0, where the contrastratio never exceed δφ = 4, shows a low contrast ratio (δφ ∼1.5) orbiting over-dense feature that produces a modulationof the accretion rate of 7–8 binary orbits.

Even though the orbiting over-dense lump maintainsa contrast ratio δφ & 10 for a limited number of orbits(∼ 1000 torb), it survives with a low contrast (δφ ≈ 1.5–2) forlonger timescales. This result does not depend on the previ-ous history of the evolution of δφ, as mentioned before forsimulation 6A3.0, where the contrast ratio never exceededδφ = 4.

These results may explain azimuthal over-dense featuresobserved in dust thermal emission in protoplanetary discs(see Sec. 5.1 for further discussion).

4.4 Effects of Different Disc Parameters

The right panels of Fig. 6 and 7 show how disc parametersaffect the eccentricity and cavity size. The relevant param-eters for the long time-scale evolution appear to be α, H/R,and q. Apart from some effects on the initial evolution allthe simulations evolve towards the same long time-scale be-haviour apart from simulation 5N (larger cavity, higher finaldisc eccentricity, H/R = 0.03 i.e. smaller than the “A” ref-erence case), simulation 5E (smaller cavity, lower final disceccentricity, α = 0.1 i.e. larger than the “A” reference case),and simulation 5C (smaller cavity, lower final disc eccentric-ity, H/R = 0.1 i.e. larger than the “A” reference case). Inthe Shakura & Sunyaev (1973) prescription the viscosity isν ∝ α(H/R)2, so the behaviour of simulations 5N, 5E and5C can be mainly attributed to the resulting differences inthe disc viscosity. However, as previously mentioned in Sec.4, the width of these resonances scales as (H/R)2/3 (Meyer-Vernet & Sicardy 1987; Teyssandier & Ogilvie 2016), so thedisc thickness may also play a role in determining the evo-lution of the binary eccentricity. This last result needs tobe supported by further studies. Simulations “Z”, that useα = 0.01 – i.e. doubled with respect to the “A” referencecase – do not appear to show significant differences fromour reference case.

5 NON-AXISYMMETRIC STRUCTURES INPROTOSTELLAR DISCS

Recent high spatial resolution observations from near-infrared to mm and cm wavelengths have revealed spiralarms (Garufi et al. 2017; Dong et al. 2018), cavities and gapsin the gas (van der Marel et al. 2016; Huang et al. 2018),

3 Miranda et al. (2017) used αss = 0.05 for the aforementionedsimulation.

and cavities and gaps in the dust (van der Marel et al. 2016;Francis & van der Marel 2020).

A small selection of the discs with cavities, display largecrescent shaped dust asymmetries (e.g. van der Marel et al.2013; Casassus et al. 2015; Tang et al. 2017b). Despite thegas distribution being directly observable using molecularlines, highest resolution images from the SPHERE instru-ment on the VLT, ALMA and other observational facilities,are sensitive to the dust thermal emission, which may differfrom the gas distribution. In particular, low contrast over-densities in the gas density structure act as pressure trapsfor dust grains (provided that they co-move with the flow),leading to perturbations in the dust density structure largerthan those in the gas.

Three possible formation mechanisms can be invokedfor these structures.

The first mechanism is a vortex co-moving with the gas(referred to as the“Vortex scenario”). For this to operate theRossby-Wave instability (RWI) must be triggered (Lovelaceet al. 1999; Li et al. 2000; Lovelace & Romanova 2014). TheRWI manifests whenever a strong gradient in the vortensityprofile is present, provided the disc viscosity is sufficientlylow (αss . 10−4). This promotes the whirling motion of ad-jacent shearing layers, similar to the Kelvin-Helmholtz in-stability. So-called “dead zones” in the disc have also beenshown to cause density gradients affecting the vortensity insuch a way that RWI is triggered (Regály et al. 2012; Rugeet al. 2016). Numerical simulations have shown that the pres-ence of a planet affects the vortensity gradient, causing somegas to accumulate outside its orbit, and carving a steep gap;for sufficiently low viscosities, the vortensity gradient is steepenough to enable the formation of a vortex.

A second formation mechanism for these dust asymme-tries is the orbiting over-dense lump discussed earlier in thiswork (Sec. 4.3). It involves the presence of a (sub-)stellarcompanion orbiting the primary star (secondary to primarymass ratio M2/M1 & 0.2), producing a poorly understoodinstability: the cavity size grows significantly and an over-dense feature orbiting at the edge of the cavity with Keple-rian motion (i.e. co-moving with the gas) forms. No whirlingmotion is observed in this scenario (Ragusa et al. 2017; Cal-cino et al. 2019).

Both the over-dense lumps and vortices are expected totrap dust particles with growing efficiency when the gas-dustcoupling is marginal (Birnstiel et al. 2013, van der Marelet al. in prep.) – i.e. when the particle Stokes number ap-proaches St ∼ 1. Indeed, both features are pressure maximaco-moving with the flow, which trap dust. However, in thevortex scenario, the growing dust-to-gas ratio within the thepressure maximum is expected to destroy the vortex (Jo-hansen et al. 2004; Fu et al. 2014), even though 3D simula-tions fail to reproduce this effect (Lyra et al. 2018).

A third mechanism to explain the origin of dust asym-metries has been discussed by Ataiee et al. (2013), the“Traf-fic jam” scenario (also discussed in this paper, Sec. 4.3). Inthis scenario the presence of a planet increases the eccentric-ity of gas orbits; the clustering of eccentric orbits and theslowdown that the gas experiences when approaching theirapocentre, causes the formation of an over-density whichis slowly precessing (not orbiting) at the same rate as thepericentre longitude of the gas orbits. This mechanism doesnot produce over-densities with high enough contrast-ratios

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18 E. Ragusa et al.

to be responsible for the formation of azimuthal over-densefeatures in discs – our simulations show that this is thecase even with cavities as eccentric as those produced by(sub)stellar companions.

Features produced by this mechanism are long lasting,as long as the disc maintain an eccentricity gradient. Butthey move with the frequency of the cavity longitude ofpericentre – in contrast to the two scenarios we previouslydescribed, where the feature spans the cavity edge with Kep-lerian frequency. Finally, eccentric features are not expectedto trap dust grains, as they are actually “traffic jams” dueto the the dust particles streamlines clustering at the apoc-entre of the orbit rather than particles being trapped in theover-density. As a consequence, the reasoning applied to thetwo previous scenarios where low amplitude gas perturba-tions could produce high contrast dust over-densities cannotbe applied in this case.

5.1 Observational implications

The most immediate observational consequence this worksuggests is that co-planar discs surrounding binaries withsufficiently high mass ratios (q ≥ 0.05) are expected to besignificantly eccentric. As soon as the disc reaches the 1:2resonance (R = 1.59 abin) the disc eccentricity grows rapidlyand the cavity size grows. This pushes material outwardsproducing an orbiting over-dense lump. In real discs, thiscondition is met in two cases; after the cloud core collapse,when the newly formed binary starts depleting the cavityarea, and when secondary accretion events take place.

Our simulations suggest that a high contrast orbitingover-dense lump (Keplerian orbit, contrast ratio δφ & 10)will last for a limited number of orbits (∼ 1000 orbits), leav-ing in its place an eccentric “traffic jam” (moving at the cav-ity pericentre precession rate, contrast ratio δφ ≈ 3–4). If thecavity is eccentric, the traffic jam feature is always present,possibly leading to the formation of two distinct azimuthalstructures: one orbiting at the cavity edge, the other fixedat the cavity apocentre.

Fig. 11 and 12 show that a low contrast gas over-density(δφ ∼ 1.5–2) is expected to survive at longer timescales – thisis consistent with previous results in the literature wherelow contrast over-densities have been observed orbiting atthe cavity edge after t ∼ 6000 torb (Miranda et al. 2017).Over-dense lumps co-moving with the flow (i.e. orbiting atthe edge of the cavity) are effective in trapping dust grainsstarting from very small azimuthal contrast ratios (δφ & 1,Birnstiel et al. 2013; Van der Marel et al., in prep); this im-plies that high contrasts in the dust distribution, as thoseobserved by ALMA, can be achieved starting from low con-trast over-densities in the gas – such as those weak orbitingfeatures that survive at late timescales in our simulations.

There are a few ways to distinguish orbiting lumps andtraffic jams observationally. First, as discussed above, sinceeccentric traffic jams cannot trap dust grains, the dust con-trast ratio in these features is not expected to exceed thecontrast in the gas. This implies that detecting azimuthalover-densities in the dust thermal emission with δφ > 3–4excludes traffic jams as a possible scenario originating thefeature. Second, the co-moving over-dense lump is expectedto trap dust, therefore detecting dust trapping (differentcontrasts δφ at different wavebands) can also help distin-

guishing the scenarios. Finally, repeated observations canbe also used. Since the over-density explored in this work isco-moving, its orbital motion can be detected (e.g. Tuthillet al. 2002). The traffic jam scenario produces a fixed fea-ture which moves on a much longer time-scale (hundreds ofbinary orbits).

Despite high contrast over-densities not lasting for timeslonger than t & 1000 torb, for binaries opening cavities aslarge acav & 100 au, thousands of binary orbits correspondto time-scales t & 105 yr, which is & 10% of typical pro-toplanetary disc lifetimes (Haisch et al. 2001; Kraus et al.2012; Harris et al. 2012). Since these features have been ob-served to form when the cavity is carved, we expect youngsystems to be more likely to show gas over-densities withδφ & 4.

Secondary accretion events, flybys, or other later per-turbations of the systems, could still in principle explain thepresence of high contrast orbiting over-dense lumps also inthe gas component of older systems.

The initial conditions in real physical systems are farfrom a steady state (Bate 2018). It is thus important tounderstand the evolution of the disc, not just when it hasreached the quasi-steady state.

Vortices differ from orbiting over-dense lumps for thefact they might form also in systems where no cavity ispresent. However, they are expected to share most of thecharacterising features of orbiting over-dense lumps – theyalso co-move with the flow and trap dust grains, causinghigh contrast dust over-densities. When a cavity is present,being able to detect the whirling motion of vortices throughkinematic maps appear to be the only strategy to distin-guish them from orbiting over-dense lumps. Nevertheless, nomolecular lines observations with the required spatial reso-lution for this purpose are available yet.

6 SUMMARY AND CONCLUSION

We performed a suite of of 3D SPH simulations of binariessurrounded by circumbinary discs. Our results suggest thatmost circumbinary discs with sufficiently high mass ratios,q ≥ 0.05, develop an eccentric cavity, consistently with previ-ous results in the literature (Farris et al. 2014; D’Orazio et al.2016; Miranda et al. 2017; Muñoz & Lithwick 2020). Theformation of eccentric cavities occurs over a wide range ofdisc+binary initial conditions, even though we always startwith a circular binary and circular disc.

Our results suggest that:

(i) The growth of the disc eccentricity appears to bedriven by an unstable positive feedback mechanism involv-ing the eccentric Lindblad resonance m = 3, l = 2, or cir-cular m = 1, l = 1 – that were previously suggested to beresponsible for the growth of the disc and binary eccentric-ity (D’Angelo et al. 2006; MacFadyen & Milosavljević 2008).However, the action of a non-resonant mechanism for disceccentricity growth cannot be fully excluded.

(ii) Despite resonances being able to explain the evolutionof the disc eccentricity, resonant binary-disc interaction the-ory alone, as we know it, seems not to be sufficient to explainthe formation of cavities as large as acav = 3.5 abin when thebinary eccentricity is as low as e . 0.01. Resonances at thoseradii are not strong enough to overcome the viscous diffusion

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of the disc. The binary eccentricity required for that purposeis much higher than the one binaries develop in this work(see Fig. 17). We speculate that ELRs increase their intrinsicstrength with increasing disc eccentricity (in addition to thealready well known and discussed dependence on the binaryeccentricity). We also speculate that a non-resonant mecha-nism might be active carving the cavity (see also point (iv)and (v) in this section, and Sec. 4.2 for more details).

(iii) The maximum value of disc eccentricity that our sim-ulations reach is set by the orbit crossing limit (see Fig. 16).This limit is not related to the maximum value of eccentric-ity, but rather the steepness of the eccentricity profile. If theeccentricity gradient (de/da) is too large, the eccentric or-bits of fluid elements intersect, resulting in shocks that pre-vent further growth of the eccentricity. In our simulations,discs around binaries with mass ratios q > 0.2 undergo or-bit intersection, reaching the same maximum value of disceccentricity. For lower binary mass ratios, viscosity acts todamp the disc eccentricity. When this effect balances thepumping action of resonances, the eccentricity stops grow-ing. Thus, for binary mass ratios q ≤ 0.2, since the intrinsicstrength of resonances grows with the mass ratio, for a fixedvalue of disc viscosity, the higher the binary mass ratio is,the higher the maximum eccentricity. This could potentiallylead to constraints on the disc viscosity for systems wherethe binary mass ratio and disc scale height can be measured.

(iv) Our analysis of disc eccentricity and cavity semi-major axis (cavity size) evidenced that these two quanti-ties show an interesting linear correlation, which appear tobe the same for all the simulations we examined (Fig. 13).We believe this result is very important, as it constitutes astarting point for future developments of this work.

(v) The pericentre radius of the cavity remains approxi-mately constant throughout the entire length of the simu-lation, suggesting that the tidal torque sets the pericentreradius of the cavity non-resonantly and, as a consequence,the cavity semi-major axis grows due to the growth of thedisc eccentricity. In the light of point iv), this points in thedirection of a non-resonant truncation mechanism being re-sponsible for carving the cavity.

Our simulations confirm some evolutionary features pre-viously observed:

(i) When the disc becomes eccentric, the material flowson nested elliptic orbits with decreasing eccentricity profileand aligned pericentres. The disc precesses rigidly, meaningthat the elliptical orbits all precess together at the same rate,conserving the alignment of the pericentres (MacFadyen &Milosavljević 2008; Teyssandier & Ogilvie 2016; Mirandaet al. 2017; Ragusa et al. 2018; Muñoz & Lithwick 2020).Eccentric cavities all show a “traffic jam” over-dense featuredue to the clustering of nested eccentric orbits at the apoc-entre.

(ii) Simulations with mass ratio q > 0.2 show the for-mation of an azimuthal over-dense feature with δφ & 10 –known in the black hole binary community as “over-denselump” and as “horseshoe” feature in the protoplanetary one(Shi et al. 2012; Farris et al. 2014; Ragusa et al. 2017; Mi-randa et al. 2017) – that orbits with Keplerian frequency atthe edge of the cavity, produced by the strong tidal streams(Shi et al. 2012; D’Orazio et al. 2013) thrown by the binary.

Our results add to this picture that the fast growth of

the cavity size appears to be one of the key ingredient forthe formation of an high (δφ & 10) contrast ratio over-denseco-moving feature. As soon as the cavity stops growing, theover-density also stops growing.

(iii) As soon as a quasi-steady state configuration isreached (after ≈ 1000 binary orbits) the disc progressivelyevolves towards a configuration with a slowly precessing “ec-centric traffic jam” feature at the apocentre of the cavityδφ ≈ 3–4 and a lower contrast orbiting over-dense lump(δφ ≈ 1.5–2) that co-moves with the flow. This result is con-sistent with what previously found by Miranda et al. (2017),who, for circular binaries, found an orbiting over-dense lumpwith δφ ∼ 2–3 is still present after 6000 binary orbits.

We draw the following conclusions for observations ofprotoplanetary discs. Both high contrast and low contrastgas structures can lead to the formation of high contrastratio features in the dust density field provided they co-movewith the flow (Birnstiel et al. 2013). This implies that oursimulations with q > 0.2 are all in principle consistent withhosting high contrast dust density structures for at least2000 binary orbits, if dust was included in our simulations.For typical systems, such timescale represents a significantfraction of their lifetime, we discuss this in Sec. 5.1. Thecontrast of this feature depends more on how much materialis pushed outward when the cavity becomes eccentric ratherthan the value of the binary mass ratio, so that q > 0.2should not be considered as a threshold for an high contrastorbiting over-dense lumps to form.

Reliable initial conditions of real physical systems arestill very poorly constrained. Given the relatively long time-scales involved, the chances of observing a system while it isstill relaxing towards a steady configuration are high, par-ticularly in young protostellar systems.

High resolution kinematic data in protoplanetary discscan be used to test our theoretical results.

Future theoretical developments of this project involvea better investigation of the strength of ELRs in eccentricdiscs, and a strategy to understand the evolution of discssurrounding high mass ratio binaries on longer time-scales.

ACKNOWLEDGEMENTS

We thank the anonymous referee for his/her insightful com-ments that substantially improved the conclusions

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